Statistical and dynamical properties of the discrete Sinai model at finite times

TL;DR

This paper investigates the statistical and dynamical behaviors of a particle diffusing in a discrete Sinai model with a reflecting boundary, revealing temperature-dependent correlations and oscillations at finite times.

Contribution

It provides explicit statistical distributions of energy minima and compares them with numerical dynamical distributions, highlighting temperature effects and finite-time behaviors.

Findings

At high temperature, distributions match asymptotically.

At low temperature, finite-time distributions show strong correlation.

Oscillatory behaviors observed in low-temperature regime.

Abstract

We study the Sinai model for the diffusion of a particle in a one dimensional quenched random energy landscape. We consider the particular case of discrete energy landscapes made of random +/- 1 jumps on the semi infinite line Z+ with a reflecting wall at the origin. We compare the statistical distribution of the successive local minima of the energy landscapes, which we derive explicitly, with the dynamical distribution of the position of the diffusing particle, which we obtain numerically. At high temperature, the two distributions match only in the large time asymptotic regime. At low temperature however, we find even at finite times a clear correspondence between the statistical and dynamical distributions, with additional interesting oscillatory behaviors.